Download Introduction to Quantum theory, and the

Introduction to Quantum theory, and the developments that lead to
Quantum Mechanics
Preface
During the course of this essay, I will be gradually going through the experimental
phenomena that lead us to a quantum interpretation of Physics, away from the
classical models from which we may begin. Quantisation is very important when
analysing these experiments and the phenomena they display as without it; we can’t
correctly predict the results of them. This essay will culminate in a roundup or
summary of the topics that we have covered and how they lead us inexorably to the
quantum mechanics which followed quantum theory. Quantum mechanics itself will
not be covered during the course of this essay.
Invention of Planck's constant and the Black body radiation phenomena
In the late 19th century physicists were looking for an explanation of the results of
experiments carried out investigating blackbody radiation. Classical derivations, like
Lord Rayleigh’s had failed to predict experimental results accurately, and so
something new needed to be tried out to help explain the phenomena. An illustration
of the relationship between the spectral emission of a blackbody and wavelength at
different temperatures is shown in the graph below.
The spectral emission S for a body with power (P) and Area (A) would be given by

S 
P
A

Therefore, the energy flux emitted in a small wavelength interval, d is S d  


Graph of spectral emission against wavelength (obtained from
http://ceos.cnes.fr:8100/cdrom-98/ceos1/science/dg/fig9.gif)
E =spectral emission
1
A black body would be a perfect absorber and emitter, and so would absorb and emit
a hundred percent of the radiation or heat it received. There is however no such thing,
and a black body is just a theoretical idea for a surface that would emit radiation, and
the radiation it emitted would be affected only by its temperature, as can be seen on
the graph, and not by its material or surface. In reality we cannot produce a
blackbody, but we can produce something that is almost like it which is a cavity. An
illustration of this is shown below.
Cavity/blackbody
As you can see from above the light ray which I have drawn as a straight line for
simplicity sake comes in and is deflected many times in the cavity. This helps
maximise the amount of absorption of the EM wave as only a small percentage of the
light is deflected each time from the material, and therefore the amount of light not
absorbed consistently decreases with every deflection. This allows the cavity to
behave almost perfectly as a radiation absorber, and effectively as a blackbody. This
means that the radiation emitted from the cavity is effectively independent of the
characteristics of the wall and only dependant on its temperature.
One thing to note is that we are assuming that the blackbody is at thermal equilibrium,
and we are also measuring its spectral emission when it is at thermal equilibrium. This
means that energy they are emitting radiation at the same rate they are receiving it,
and so there is no energy gradient overall; between the radiated energy and the energy
in the walls of the cavity. If the radiation absorbed by the atoms has to be the same as
that emitted by them, we must therefore have standing waves between the walls of the
cavity, with amplitude of zero at the walls. If this were not the case then energy would
be dissipated at the walls and violate our assumption of equilibrium. This would mean
that we could have a greater number of possible standing EM waves for smaller
wavelength EM waves than for larger wavelength resonant EM waves. This is
illustrated in the diagrams on the next page:
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High frequency and low frequency standing EM waves in cavity
As can be seen from the diagram above, more of the higher frequency standing waves
can fit in the cavity. By taking this into account Lord Rayleigh produced a formula for
the number of EM waves per unit volume per unit wavelength that could fit in a
blackbody. From this it would be possible for us to work out the energy density per
unit volume per unit wavelength. It was here that Lord Rayleigh and James Jeans
made an assumption that turned out to be inaccurate. They assumed that the
equipartition of energy applied to the standing waves, and so each EM wave would
have the same average energy kT. Therefore:
 E   kT
This would also mean that the energy of each standing wave would be irrespective of
its frequency. The formula for the energy density per unit wavelength is shown below,
and I have circled the part representing the assumed average energy of each standing
wave kT. The other part of this formula represents the number of possible standing
waves per unit volume per unit wavelength.
dE 8 kT

d
4
E=Energy density ( Jm
K=Boltzmann constant
3
)
(1.3806503 × 10-23 JK 1 )
This formula however predicts the propagation of a very large number of high
frequency EM waves leading to very high predicted values for the energy density of
the EM waves emitted by the blackbody. This is nowadays referred to as the
“ultraviolet catastrophe,” which is what would occur basically if nature behaved this
way. For example even the heat from a small fire would produce very large amounts
life threatening radiation [in other words a catastrophe]. This can be seen from the
graph on the next page.
3
spectral emission(Wm^-3)
Ultraviolet catastrophe(spectral emission vs
wavelength)(calculated using Rayleigh's formula
with several wavelngth values)
2.50E-10
High frequency radiation
has a spectral emission
tending to infinity.
2.00E-10
1.50E-10
1.00E-10
5.00E-11
0.00E+00
0.00E+0 2.00E0
08
4.00E- 6.00E- 8.00E08 wavelength(m)
08
08
1.00E07
1.20E07
This is where Planck took a radical departure from the classical physics employed in
the Raleigh Jeans formula, while keeping the part of the formula that described the
number of possible standing waves per unit volume, per unit wavelength, which was
still correct. Planck proposed that the energy of what he referred to as oscillators [the
atoms] could only exist as multiple of hv. Therefore the energy of the oscillators in
the black body could only be emitted in discrete lumps or quanta. By appreciating
this, we can understand the components used in Planck’s formula, which are not
present in Rayleigh’s.:
H=Planck’s constant ( 6.626068 × 10Js-1 )
E p  Energy per standing wave of a
Ep  
hv
e
hv / kT
1
particular frequency multiplied by its
probability of occurring (J)
This equation can be understood by splitting it into two parts. The first part which is
circled represents the energy of a quantum or packet of energy (of light). The rest
represents the probability that the energy of the quantum will be occupied. As can be
seen from inspection, if we increase the frequency of the light we are observing; the
likelihood that we observe it drops if temperature remains constant. Also a higher
frequency photon would have a higher energy. It also reveals that the energy is
hc
quantised and comes in packets of energy hv or .

By replacing kT with the expression above in Rayleigh’s formula for the energy
density per unit wavelength of; we get Planck’s equation for the energy density of a
standing EM wave in a resonant cavity, per unit wavelength.
dE 8
hc
 5 hc /  kT
d  e
1
This turns out to be the correct expression for the energy density per unit wavelength
of emitted radiation. It happens that this equation almost exactly describes the nature
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of EM waves that we observe, and reveals to us that energy critically is quantized, and
not continuous. We can now compare both these formulae.
According to
Rayleigh:
According to Planck:
dE 8 hc 1

d   5 ehv / kt  1
dE 8 kT

d
4
From both of these we can find the spectral emission (power per unit area per unit
wavelength) predicted by Rayleigh and Planck respectively. To find this we multiply
c
the above expressions by .
4
Spectral emission  S
According to Planck:
According to Rayleigh:
2
S 
2 ckT
S 
4
2 c h 1
 5 ehv / kt  1
From both of the above formulas we can find spectral emission, but only with
Planck’s formula, which takes into account the energy of a photon being:
E  hv
Or
E
hc

Another key difference in Planck’s formula was he took into account the probability
that a photon of a particular frequency would be emitted at a certain temperature. This
was:
Probability 
1
e hv / kt  1
Using Planck’s formula, we can very accurately predict our results comparing spectral
emission, displaying one of the triumphs that the quantum theory brought us and how
it allowed us a better view into how light and energy existed on a small scale, and
how this affects us. A comparative graph of both curves produced by the Planck and
Rayleigh formulas is displayed on the next page.
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Graph of Planck and Rayleigh radiation curves. Graphs
show spectral emission against wavelength.
5E+13
1.2E+18
4E+13
1E+18
3E+13
8E+17
6E+17
2E+13
4E+17
1E+13
Planck Spectral Emission (W/m3)
Rayleigh Spectral Emission (W/m3)
1.4E+18
2E+17
0
0
100
200
300
400
500
600
700
800
900
1000
Wavelength (nm)
The photo-electric effect
The photo-electric effect was the next experimental phenomena, which was explained
using the principle of quantisation by Einstein, after the Planck black body radiation.
EM radiation striking a metal surface (obtained from
http://www.llnl.gov/str/June05/gifs/Aufderheide3.jpg)
As shown above; it comprised of the effect first observed by Heinrich Hertz, whereby
when EM radiation is shone on a metal surface, it causes the ejection of electrons
from the surface. This on its own could be explained by a classical wave
interpretation of light, were it not for a few observations that were made when
observing this effect that contradicted such an interpretation.
 Emission of electrons only occurred if the frequency of the radiation was above
a minimum value, known as the threshold frequency that varied depending on
the metal being irradiated.
 Emission of electrons began as soon as a surface was irradiated.
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 If the EM radiation has a frequency more than the threshold frequency, then the
No of photons emitted was proportional to the intensity of the radiation.
 The kinetic energies of the ejected electrons were proportional to the frequency
of the EM radiation, although increasing the intensity of the light had no effect
on the KE of the electrons.
According to the wave interpretation, the energy of the light should be uniformly
distributed over a wave front. Therefore, each electron on the metal surface should be
able to absorb an equal amount of radiant energy when irradiated. If this were the
case, then if the intensity of the light were low, no ejection of electrons would be
observed, or a certain amount of time at least should pass until an electron has
acquired enough energy to be emitted. However, as can be seen experimental
observations contradicted these assumptions. Also if the light did behave as wave, by
increasing the intensity of the light, the energies of the electrons emitted should have
also increased. This also did not occur.
To explain this phenomenon, Einstein started off by using Planck’s equation for the
energy a quanta or packet of radiant light. Planck had used this idea to suggest that the
light could only be emitted in discrete packets of energy.
E  hv
Einstein extended this idea to suggest that upon emission; the light continued to exist
as indivisible packets or particles of energy (or photons). A light beam could therefore
be considered to be a stream of photons. This idea of particles, would then be
followed by the principle that only one electron could absorb the energy of one
photon. This means that it would only take a single high energy photon to begin
photoemission, and so explains why emission of electrons begins as soon as a surface
is irradiated.
There will however need to be a certain amount of energy that will need to be
provided, to overcome the force binding the electron to the metal surface. This
amount of energy is known as the work function W and it varies depending on the
metal used. Also, if a photon, without the required energy hit an electron, it would be
reflected, and the electron would not accumulate energy.
If a photon strikes a metal surface, causing the emission of an electron then:
E  hv  W 

1
2
mv 2

max
W=work function of metal (J)

1
2
mv 2

max
=maximium KE of electron
upon emission
=Energy of photon causing emission
From this it can be seen that there will be a minimum frequency at which emission
occurs. To emit an electron: hv  W Therefore if v0 is the minimum, or threshold
frequency: hv0  W
It is important to note that this formula only works with electrons that have the
maximium amount of KE and therefore have not dissipated any of their initial KE
after emission. Also, if a more tightly bound electron in a metal with a greater work
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function is removed by a particular photon; that electron will have a smaller KE than
an electron removed by the same photon, from a metal with a smaller work function.
This equation that Einstein developed was rigorously tested by Robert A Millikan in a
series of refined experiments in 1916. The results of these experiments completely
verified Einstein’s theory.
Cathode
e
V
Anode
A
Experiment to test photo-electric effect
The above diagram is not an illustration of Millikan’s experiment but of a similar,
simple experiment that can be carried out in order to test Einstein’s equation. Notice
that the cell has been reversed, so that a negative PD is applied at the anode and vice
versa. This is so that electrons emitted from the cathode, which is irradiated with UV
light, are repelled by the anode. Because of this we can measure the KE of the
electrons by making use of the following relationship:
eV 
1 2
mv
2
Therefore, according to Einstein: eV
 h(v  v0 )
A force acts to oppose the motion of the electrons towards the anode and only a few
have sufficient energy to overcome this, allowing a small current to pass. We can
measure their max KE by gradually increasing the PD up to a point at which the
current is stopped altogether. By taking a reading of the PD at this point we can find
the Max KE of the electrons for a particular frequency. If Einstein was right, we
should be able to get a linear relationship between the stopping PD and the frequency
of the light. This is indeed what is observed, and the relationship is illustrated in the
graph below. Planck’s constant can also be found by finding the gradient of the graph
below and dividing it by the charge on an electron.
8
Threshold frequency of metal
Graph of stopping PD against frequency (obtained from
http://physics.rgsguildford.co.uk/Physics/Photoelectric%20effect/Resultados.gif
The experimental phenomena we have now been through show us that energy is
quantized at the fundamental level and is emitted and absorbed in discrete quanta,
rather than being infinitely separable and divisible, for example; it does not exist
uniformly distributed in an EM wave, but instead exists in the form of individual
photons.
Spectral lines and Bohr’s theory of the atom
The next major problem that was unexplainable by the physics of the time, but which
quantisation helped explain was the spectral lines of hydrogen and a formulation of
the structure of the atom by Niels Bohr. This was after experimental work by
Rutherford and his colleagues proving that the majority of the mass and all the
positive charge in an atom subsided in the nucleus. The electrons were assumed to
orbit the nucleus, but there was no reasonable explanation for why the electrons
would not hurtle towards the atom, emitting an increasing amount of radiation along
the way.
When a gas is heated in a container, and made hot enough to emit visible light which
then passes through a thin slit, before being refracted by a triangular prism wedge; it
produces a bright line spectra. A top down diagram of this experiment is shown on the
next page.
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A similar experiment can be used to find absorption spectra, by placing a container of
Container of
glowing hot gas
Narrow slit
Emitted light
Prism
Screen displaying spectral
lines
cool gas between the narrow slit and a bulb emitting radiation. When this is done the
continuous spectrum produced by the bulb, is broken by dark spectral lines. An
interesting property of these dark spectral lines is they appear in the same places as
the bright spectral lines, as well as the fact that they are different for every element. It
can be considered, therefore that the nature of these spectral lines, and the type of
radiation absorbed and emitted by different elements; is dependant on their atomic
structure.
In order to relate the line spectra to a tangible theory of atomic structure, physicists at
the time chose to investigate the spectra of hydrogen, as it is the simplest element.
Upon careful mathematical analysis of the experiments carried out in this field,
Johann Jakob Balmer was able to obtain a formula which almost exactly predicted the
frequencies of the four visible spectral lines, and used it to predict the frequencies of
others in the UV and infrared ranges. These were later confirmed to a near exact
degree of accuracy.
R=Rydberg constant (1096.78m-1 )
 1
1 
ni =initial principal quantum number
v  cR  2  2 
n
n f =final principal quantum number
n 

f
i

The values of ni and n f were actually found to be principal quantum numbers of the
atoms of hydrogen later on. At the time when Balmer discovered his formula he found
that if the final number was 2 and the initial numbers were 3, 4 or 5; he was able to
produce the frequencies of the light producing the visible spectral lines. Balmer’s
formula, turns out to be critical to atomic theory because it relies on changes of whole
numbers, it suggests that changes in the internal structure of the atoms are involved in
producing the emissions of EM light. The Balmer formula itself however, can reveal
to us no information about the internal structure of an atom.
In producing his model of the atom, Bohr took into account the Balmer formula, the
ideas produced by Einstein and Planck, regarding energy and light only to be emitted
and absorbed in lumps or quanta.
10
When formulating his model of the atom, Bohr came up with several postulates.
These were that:
 The orbits and energies of the electrons were quantized, meaning that only
specific orbits and energies could be occupied by electrons. When an electron
would be in one of these quantized orbits it would be in a stationary state and
not accelerate towards the nucleus and emit any EM radiation. The electron
could however move discontinuously between different orbits stationary states.
During a transition between a higher stationary state and a lower stationary
state for example; the electron would emit radiation.
 When the electron makes the transition between the 2 energy states, a photon
would be emitted by the electron. The frequency of this photon would depend
on the energy difference between the 2 states according to the formula:
v
E
h
E =Energy difference between 2 stationary states
 The laws of classical physics applied to the orbits of the electrons in a
stationary state, but not when considering their transitions between different
states.
 The orbital angular momentum L of the electrons is quantized too, and can only
h
have values that are an integral multiple of
. Therefore:
2
L
nh
2
n=1, 2, 3…
From the basis of these postulates, it is possible for us to work out the stationary
states and the spectrum of the hydrogen atom, as long as we consider for
simplicity sake; that the electron is moving in a circular orbit around a stationary
proton. This would mean that:
nh
 me vr
2
v=velocity of particle (ms 1 )
r=radius of orbit (m)
me =mass of electron (kg)
First of all we can consider the electrostatic force F of attracting the electron to the
proton. For simplicity sake, we also consider the nucleus to be a point charge
e=charge on an electron
e2
0 =permittivity of free space
F
4 0 r 2
( 8.85 1012 C 2 N 1m2 )
Therefore, as the centripetal acceleration of an object is given by
second law:
e2
mv 2

2
4 0 r
r
11
v2
, by Newton’s
r
Therefore:
me2 r
 m2 v 2 r 2  (mvr ) 2
4 0
As
nh
mvr 
2
(n=1, 2, 3…)
me2 r  nh 


4 0  2 
2
We can now work out the radius of the orbit of an electron by rearranging the
formula above to give:
2 2
0
2
e
The radius of the smallest orbit r0 would be given by when n=1 so therefore:
r
nh 
m
h2 0
11
r0 

5.29

10
m(3sf )
2
 me
All of the other electron orbit radii of hydrogen are multiples of this value by the
formula:
rn  n 2 r0
The energy of an electron in one of these orbits would be the sum of its kinetic
energy Ek and electrostatic potential energy E p . Therefore:
E  Ek  E p
2
e
E  12 mv 
4 0 r
2
The potential energy is considered negative because the nucleus is attracting the
electron, and work has to be done against it in order for the electron to escape the
atom. We can rewrite this equation as we can put the KE of the electron in terms
of the potential energy, as can be seen below, by making use of the relationship
shown on the bottom of the previous page.
2
2
1
2
0
0
E
e
e

4  r 4  r
Therefore
12
e2
E
8 0 r
n 2 h 2 0
r
As:
 me 2
Therefore
me 4
E
8 0 2 n 2 h 2
We can use the above equation now to predict the energies of the various energy
levels of the hydrogen atom that are displayed in the diagram below.
Infrared
light
Visible
light
UV
light
Energy levels of hydrogen and the spectral lines produced by it (obtained
from http://www.daviddarling.info/images/hydrogen_spectrum.gif).
For example, the energy required to remove an electron with the lowest quantum
number 1, in other words at the ground state; is:
4
18
e
2 2
0
We can now find the energy of the photon emitted, or the change in energy E when
an electron drops from an initial energy state to its final energy state.
m
E
 2.18 10
8 h
13
J  13.6eV
E  Ei  E f


me 4
me 4
E  

2 2
2 2 

8 0 ni h  8 0 n f h 
me4  1
1 
E 
 2
2 
2

8 0 h  n f
ni 
As you can see, our formula is starting to look strikingly similar to Balmer’s. By
dividing both sides by Planck’s constant, we can get the expression now for the
frequency of the emitted photon.
me 4  1
1 
v
 2
3 
2

8 0 h  n f
ni 
You may now notice that all of the terms outside the brackets are physical constants,
and if we compute the calculation outside of the brackets we get a theoretical value
very close to the experimentally determined Rydberg constant.
(Note: This calculation only includes the principal quantum numbers and energy
levels. Soon after Bohr’s initial formation of his theory, many other quantum numbers
like the magnetic were to follow to produce a better picture of the hydrogen atom,
improving the predictions made by it. Further refinement was also made by Wolfgang
Pauli with his exclusion principle, not allowing 2 electrons to occupy the same state,
causing them to spin in opposite directions)
De Broglie hypothesis of matter waves
Another important concept that was developed for Quantum theory was that of matter
waves first suggested by Louis de Broglie. As we had found out earlier, that light
which was previously considered consensually as a wave could also behave and be
considered as a particle, and really was neither. We could equally however consider
electrons and other particles as waves.
For example I have placed below both equations of Energy from relativity and the
photo electric effect. By simple algebraic manipulation of these equations we can find
an expression for the wavelength of a photon in terms of its momentum and Planck’s
constant.
2 2
2 4
E

p
c

m
By relativity:
0 c
When considering a photon of zero rest mass however:
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E 2  p 2c 2 Therefore: p 
E
c
p= momentum  kgms 1 
Considering the photo-electric effect:
E  hv 
hc

E
h

c


Therefore for a photon:
h
p
The question de Broglie asked was; could this apply for other phenomena apart from
photons like electrons for example. As we know the mass of electrons, and for a
particular scenario, its velocity, if we considered it a wave we could find its
wavelength by the following equation:  
h
mv
As was later confirmed by GP Thompson, electrons can also behave like waves, and
in fact all objects have de Broglie wavelengths. It is important however to note that
these are not “real” waves in reality but probabilistic ones. The regions of the waves
with high amplitude indicate a high probability of finding a particle in that location.
Another thing to note is that the wavelength of the de Broglie wave becomes smaller
as the object in question has a larger momentum. This means that we do not
experience this de Broglie wave with the everyday objects we encounter.
De Broglie’s hypothesis was important in relation to Bohr’s model of the atom, as
electron orbits could also be considered to be standing EM waves and this is in fact a
way to derive Bohr’s expression for quantized angular momentum of an electron. For
example, if we consider the perimeter of an electron’s orbit to be equivalent to a
number of standing de Broglie waves, we obtain the following relationship:
2 r  n
Therefore,
2 r 
nh
p
And so;
rp 
nh
 mvr
2
De Broglie’s ideas would turn out to be very important for the development of
Quantum mechanics (for example when considering the Schrödinger wave equation)
after the initial steps made in developing what I have labelled during this essay as
Quantum theory.
Summary
The Bohr model of the atoms in many ways was a culmination of the early ideas
preceding the development of a full quantum theory. It expanded on the ideas
developed by Planck and Einstein concerning quanta and related this to how it
15
affected atoms and was determined by atomic structure. It was able to explain the
structure in the spectrum of hydrogen in terms of the atomic structure of hydrogen.
However the Bohr model should be understood as the first step towards giving us a
realistic understanding of atomic structure and the quantum world, as it suffered from
various flaws and shortcomings so; was nowhere near a complete theory.
This would have to wait until the development of Quantum mechanics. The principle
of quantisation of energy, developed by Planck and Einstein was only the first step in
a journey that would end up with a drastically different picture of the physical world
at its fundamental level, from the one with which it started. De Broglie’s ideas about
matter waves turned out to be a critical analogy when considering the Schrödinger
equation and the probabilistic wave function of quanta It is important also to
understand that Bohr’s model of the atom was a hybrid theory, combining aspects of
classical and quantum physics, and only a full explanation in terms of quantum
physics could properly account for atomic phenomena.. Also Bohr’s original
postulates could be justified by the applying the quantum mechanics which followed
his theory, to the atom.
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